Well posedness and stochastic derivation of a diffusion-growth-fragmentation equation in a chemostat

TL;DR

This paper establishes the existence and uniqueness of solutions for a complex diffusion-growth-fragmentation model in a chemostat, derived from stochastic individual-based models, and explores the regularity of the solution.

Contribution

It provides new mathematical results on well-posedness and stochastic derivation of a nonlinear coupled system modeling bacterial growth in a chemostat.

Findings

Proved existence and uniqueness of solutions under various conditions.

Showed the stochastic trait dynamics have a density with Besov regularity.

Connected measure solutions to probabilistic properties of the stochastic model.

Abstract

We study the existence and uniqueness of the solution of a non-linear coupled system constituted of a degenerate diffusion-growth-fragmentation equation and a differential equation, resulting from the modeling of bacterial growth in a chemostat. This system is derived, in a large population approximation, from a stochastic individual-based model where each individual is characterized by a non-negative real valued trait described by a diffusion. Two uniqueness results are highlighted. They differ in their hypotheses related to the influence of the resource on individual trait dynamics, the main difficulty being the non-linearity due to this dependence and the degeneracy of the diffusion coefficient. Further we show that the semi-group of the stochastic trait dynamics admits a density by probabilistic arguments, that allows the measure solution of the diffusiongrowth-fragmentation equation to be a function with a certain Besov regularity.